First, forgive me for resurrecting an old post. But this has bothered me long enough.
It seems to me that wi-fi is indeed used for MMS texts. This is based on a simple observation that my MMS texts send/receive MUCH faster when I'm connected to wi-fi, as compared to my 3G connection when wi-fi is off. My old Android (4.1.something) used to NOT dl MMS texts when I was on wi-fi at work (I had to manually turn off wi-fi and force a dl through 3g/4g), but would have no problem downloading them when I was on my home wi-fi. If I had a data connection at both places, why would it not automatically download them at work? Then suddenly, MMS at work started to work. I don't believe there was an update to my phone, so I suspect that the 'open' wi-fi network at work had some settings changed to be more 'open'. My new Moto G is 3G only, and is slow for data. Sending/receiving MMS texts is a noticeable process when I'm not on wi-fi. When I am on wi-fi, it's almost instantaneous.
I'm open to being educated on this, so please correct me if I've drawn false conclusions.